"I'll not budge an inch" is the greatest contribution the tinker
Christopher Sly has made to everyday English. A version of the
phrase—"I'll not yield an inch"—had entered the language not long
before the composition of The Taming of the Shrew, but Sly's
particular spin has endured. The use of "budge" as a verb, meaning
"stir, move," was also very new; the earliest example we have is
from 1590, about three years before Shakespeare's play.

As the OED points out, the verb has always been used
almost exclusively in negative phrases—you will always "not
budge" from somewhere; to say "Oh, sure, I'll budge if you like"
sounds absurd. In another comedy written not long after
Shrew, however, Launcelot Gobbo reports a contest between
his good and bad consciences, the former saying "Launcelot, bouge
not," while the latter urges him to "bouge" (The Merchant of
Venice, Act 2, scene 2).

As The Taming of the Shrew opens, Sly stumbles drunkenly out of
an alehouse, where he's been a rowdy fellow. When he spurns the
hostess's demand that he pay for the glasses he's broken, she goes
to fetch the law—the "thirdborough," or constable. His nonsensical
response—"Go by, Saint Jeronimy!", etc.—mangles a famous line from
Thomas Kyd's The Spanish Tragedy ("Go by, Hieronymo!"). Sly
is one of those playgoing apprentices who, as London magistrates
often complained, indulged in the corruptions of the theater and
drunken carousing when they ought to have been tinkering at the
shop.